The total angular defect is the sum of the angular defects over all polyhedron vertices of a polyhedron, where the angular defect at a given polyhedron vertex is the difference between the sum of face angles and . For any convex polyhedron, the Descartes total angular defect is
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(1)
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This is equivalent to the polyhedral formula for a closed rectilinear surface, which satisfies
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(2)
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A polyhedron with equivalent polyhedron vertices is called a Platonic solid and can be assigned a Schläfli symbol . It then satisfies
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(3)
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and
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(4)
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so
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(5)
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